Question 291299
This IS a nasty problem. It can be solved. I just have to grind it out (WITHOUT A CALCULATOR)! Let's go!
I'm gonna go fast, so hold on!
we have x(x+1)=2162
x^2+x-2162=0
We know that a product with the difference of 1=2162.
Where do we start? The numbers are close, so how about trying to figure
out close squares? 21 x 100=2100 pretty close --10 x square root of 21.
square root of 21 is between 4 and 5. 10 x 4 is 40! square this, we get 1600.
What two numbers between 40 and 50 (2500 is too large) are consecutive whose products in the ones' column end in a 2?  try 43 and 44---1892, too small       48 and 49---2352, too large
46 and 47---2162---BINGO

IMORTANT--I DON'T HAVE A CALCULATOR!!!!    YOU CAN SHOOT THIS THRU THE QUAD AND YOU END UP WITH LOOKING FOR THE SQUARE ROOT OF 8649. WITH A CALCULATOR THIS IS EASY, BUT BY PLUGGING AWAY AND NARROWING DOWN THE PRODUCTS OF 2162 (FROM INITIALLY EXTRAPOLATING THE SQUARE ROOT ACTUALLY) WE GET THE ANSWER. THE WAY I DID THIS IS PROBABLY NOT THE WAY YOUR TEACHER WANTS YOU TO SOLVE IT. BUT GO OVER WHAT I DID. I ASKED MYSELF THE QUESTION, WHAT TWO NICE SQUARES OF NUMBERS DOES 2162 FALL BETWEEN--40 AND 50. AND, WHAT NUMBERS BETWEEN 40 AND 50 WHEN MULTIPLIED TOGETHER WILL YIELD A PRODUCT ENDING WITH THE NUMBER TWO--2 IN THE ONES' COLUMN